This paper aims to evaluate the impact interaction between the abutment and the girder subjected to nonuniform seismic excitation. An impact model based on tests is presented by taking material properties of the backfill of the abutment into consideration. The conditional simulation is performed to investigate the spatial variation of earthquake ground motions. A twospan continuous steel box girder bridge is taken as the example to analyze and assess the pounding interaction between the abutment and the girder. The detailed nonlinear finite element (FE) model is established and the steel girder and the reinforced concrete piers are modeled by nonlinear fiber elements. The pounding element of the abutment is simulated by using a trilinear compression gap element. The elasticperfectly plastic element is used to model the nonlinear rubber bearings. The comparisons of the pounding forces, the shear forces of the nonlinear bearings, the moments of reinforced concrete piers, and the axial pounding stresses of the steel girder are studied. The made observations indicate that the nonuniform excitation for multisupport bridge is imperative in the analysis and evaluation of the pounding effects of the bridges.
Long span bridges are subjected to environmental loadings and dynamic excitations due to the interaction between the bridges and the surrounding environment [
Prior to conducting the impact analysis, a rational impact model is essential to reflect the physical collision scenario. There exist many pounding models which have been presented to account for the impact effect between adjacent structures in the past several decades. Muthukumar and DesRoches [
The spatial variations include three aspects [
The primary objective of this study is to investigate the pounding interaction between the abutment and the steel girder subjected to the uniform and nonuniform excitations with different wave propagation speeds. An elasticperfectly plastic element is used to simulate the nonlinear rubber bearings. A trilinear impact model based on experimental data is adopted to simulate the pounding effects between the abutment and the girder. The FE model is established in the package Open System for Earthquake Engineering Simulation (OpenSees). The steel girder and the reinforced concrete piers are modeled by nonlinear displacementbased fiber elements. The nonstationary conditional simulation is performed to generate the asynchronous ground motions. A real twospan continuous steel box girder bridge is taken as the example to analyze and assess the pounding interaction between the abutment and the girder. The comparisons of the pounding forces, the shear forces of nonlinear bearings, the moments of reinforced concrete piers, and axial pounding stresses of the steel girder are studied. The made observations indicate that the nonuniform excitation for multisupport bridge is imperative in the analysis and evaluation of the pounding effects of the highway bridges for it always results in disadvantageous load case.
Several impact models have been presented by many researchers [
Forcedeflection relationship of the abutments.
The outofphase effect on the pounding of adjacent girder and the abutment is mainly investigated in this paper. Therefore, as one of the primary factors resulting in the inhomogeneous phenomenon at different supports of the bridge, the spatial variation of seismic waves is an imperative issue required to be solved. Three factors may affect the spatial variations of seismic ground motion, the wave passage effect, the incoherence effect, and the local site effect [
Let us consider a segment of the ground motion at a point
Consider the simulation of seismic ground motions at a set of
The remaining Fourier coefficients at frequency
Flowchart of conditional simulation.
For a lumped mass system, the dynamic equilibrium equation in terms of the nonuniform excitation can be written as
It is noted that the pounding scenario which generates high magnitude and short duration acceleration pulse during an earthquake will make the numerical convergence difficult. Therefore, a variable time stepping procedure is utilized to determine the impact time and solve the equation of motion.
To assess the pounding interaction between the girder and the abutment subjected to the nonuniform seismic excitations, a real twospan continuous steel girder bridge constructed in China is selected as an example. The bridge is supported on reinforced concrete piers and the two spans are 50 m and 70 m, respectively, as shown in Figure
Bearing properties.
Location of bearings 




Yielding strength (kN) 

Left abutment  1.0 
0.03  0.15  209.0  20.9 
Middle pier cap  1.0 
0.126  0.15  838.0  83.8 
Right abutment  1.0 
0.03  0.15  209.0  20.9 
Profile of a steel continuous girder bridge.
Profile of a twospan steel continuous girder bridge (unit: cm)
Detailed cross section of the pier and abutment (unit: cm)
Detailed cross section of a pier (unit: mm)
The FE model of this bridge is established with the aids of the package OpenSees. The steel box girder and the reinforced concrete pier are modeled by using nonlinear fiber elements. The rubber bearings are modeled by using an elasticperfectly plastic zerolength element. A trilinear zerolength element is used to simulate the impact effects between the abutment and the girder. The dynamic equilibrium equation involved multisupport excitation which can be solved using the Newmarkbeta method with a selfadaptive integration time step from 1.0
To evaluate the pounding interaction between the girder and the abutment of the continuous steel girder bridge, the El Centro NS (1940) ground motion is selected as inputs to the example bridge structure. The time history of the seismic record is shown in Figure
Original time histories of El Centro (NS).
Dynamic responses of abutments (
Left abutment
Right abutment
Dynamic responses of the middle foundation (
Dynamic responses of abutments (
Dynamic responses of the middle foundation (
As discussed above, three scenarios are considered to evaluate the dynamic responses of pounding interaction between the abutment and the steel girder. Figure
Peak response of pounding force.
Location  Uniform  Nonuniform (400 m/s)  Nonuniform (200 m/s)  

Time (s)  Force (kN)  Time (s)  Force (kN)  Time (s)  Force (kN)  
Left abutment  15.46  2.79 
7.96  2.79 
12.12  1.57 
Right abutment  14.88  2.79 
7.52  2.79 
12.62  2.79 
Time history of pounding force at abutments.
Left abutment
Right abutment
Figure
Peak responses of shear forces.
Location  Uniform  Nonuniform (400 m/s)  Nonuniform (200 m/s)  

Time (s)  Force (kN)  Time (s)  Force (kN)  Time (s)  Force (kN)  
Left bearing  3.74  41.9  4.7  41.9  8.2  41.9 
Middle bearing  2.34  168.0  3.02  168.0  4.02  168.0 
Right bearing  3.74  41.9  5.08  41.9  7.10  41.9 
Time history of shear force of bearings.
Left bearing
Right bearing
Middle bearing
Figure
Peak response of moment force and curvature.
Case  Curvature (1/m)  Moment (kN·m) 

Uniform  0.0026  7.51 
Nonuniform (400 m/s)  0.0051  9.28 
Nonuniform (200 m/s)  0.0032  8.06 
Time histories of moment force at the bottom of RC pier.
Moment versus curvature relationship of piers.
To be the primary component of the bridge, the steel girder bears the moments induced by various loadings, such as dead load, live load, and temperature load. If the pounding interaction between the girder and the abutment occurs during an earthquake, the axial stress in the girder may increase sharply to induce damage events of the steel girder. The time histories of axial stress at both ends of the girder are computed and displayed in Figure
Peak response of axial stress of steel girder.
Location  Uniform  Nonuniform (400 m/s)  Nonuniform (200 m/s)  

Time (s)  Stress (kPa)  Time (s)  Stress (kPa)  Time (s)  Stress (kPa)  
Left end  15.28  8.26 
7.78  8.39 
11.92  4.44 
Right end  16.0  8.05 
7.34  8.96 
12.44  7.97 
Time history response of axial stress in steel girder.
Left end of girder
Right end of girder
The axial stress responses of the steel girder indicate that the most dangerous loading case for both ends of the steel girder is the nonuniform excitation with the wave propagation speed of 400 m/s. However, the nonuniform excitation with the wave propagation speed of 200 m/s presents the smallest seismic responses.
The paper investigates the pounding effects between the abutment and the steel girder by considering the uniform and nonuniform excitation with different wave propagation speeds. A detailed nonlinear FE model is established and two types of zerolength nonlinear elements are used to model the pounding interaction between the girder and the abutment and the rubber bearings, respectively. In addition, two types of nonlinear fiber displacementbased elements are used to model the steel box girder and the reinforced concrete pier, respectively. The dynamic analyses on impact effects are carried out under the uniform and nonuniform excitations with two wave propagation speeds. The comparison of the seismic responses, such as the pounding force between the abutment and the girder, the shear forces of the rubber bearings, the moments of reinforced concrete pier, and the axial pounding stresses in the steel girder is performed.
The consequence of the comparison indicates that the nonuniform excitation always results in the disadvantageous responses in the seismic analysis of the bridge under impact action.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful for the financial support from the National Natural Science Foundation of China (51178366), the Technological Project of the Chinese Southern Power Grid Co. Ltd. (KGD20130222), the Fok YingTong Education Foundation (131072), and the Natural Science Foundation of Hubei Province (2014CFA026).